Fourier Transform of Curl and Divergence Free Vector Function

[LocationX]

For some reason I can't post everything at once... gives me an error
A vector function can be decomposed to form a curl free and divergence
free parts:

$$\vec{f}(\vec{r})=\vec{f_{\parallel}}(\vec{r'})+\vec{f_{\perp}}(\vec{r'})$$

where

$$\vec{f_{\parallel}}(\vec{r'}) = - \vec{\nabla} \left( \frac{1}{4 \pi} \int d^3 r' \frac{\vec{\nabla'} \cdot \vec{f}(\vec{r'})}{|\vec{r}-\vec{r'}|} \right)$$

and

$$\vec{f_{\perp}}(\vec{r'}) = \vec{\nabla} \times \left( \frac{1}{4 \pi} \int d^3 r' \frac{\vec{\nabla'} \times \vec{f}(\vev{r'})}{|\vec{r}-\vec{r'}|}$$

I am trying to take the Fourier transform of $\vec{f_{\parallel}}(\vec{r'})$ and $\vec{f_{\perp}}(\vec{r})$


I am starting at $\vec{f_{\parallel}}(\vec{r'})$. We know that the Fourier transform is given by:

$$\vec{f}(\vec{k}) = \int_{-\infty}^{\infty} d^3r e^{- i \vec{k} \cdot \vec{r}} \vec{f}(\vec{r})$$


$$\vec{f}(\vec{r}) = \frac{1}{(2 \pi)^3} \int_{-\infty}^{\infty} d^3k e^{- i \vec{k} \cdot \vec{r}} \vec{f}(\vec{k})$$


I'm not exactly sure where to begin. If I just plug and chug , we'd have:

$$\vec{f}(\vec{k}) = \int_{-\infty}^{\infty} d^3r e^{- i \vec{k} \cdot \vec{r}} \vec{f}(\vec{r})$$

$$\vec{f}(\vec{k}) = \int_{-\infty}^{\infty} e^{- i \vec{k} \cdot \vec{r}} - \vec{\nabla} \left( \frac{1}{4 \pi} \int \frac{\vec{\nabla'} \cdot \vec{f}(\vec{r'})}{|\vec{r}-\vec{r'}|} d^3 r' \right) d^3r$$


I just do not see a simple way of tacking this problem. Any thoughts would be appreciated.

 

[Jhero]

Thank you for sharing your question and thoughts on this topic. I can understand your frustration with not being able to post everything at once. Technical errors can be quite frustrating, but it's important to stay calm and approach the problem with a clear mind.

In regards to your question, taking the Fourier transform of vector functions can be a tricky task, but it is definitely doable. Your approach so far is on the right track, but there are a few steps that need to be clarified.

Firstly, when taking the Fourier transform of a vector function, we need to consider the individual components of the vector separately. This means that we need to take the Fourier transform of each component of \vec{f_{\parallel}}(\vec{r'}) and \vec{f_{\perp}}(\vec{r'}).

Secondly, we need to keep in mind that the Fourier transform of a derivative is not simply the derivative of the Fourier transform. In this case, we need to use the properties of the Fourier transform to simplify the integral before taking the derivative.

Finally, after taking the Fourier transform of each component, we can combine them to get the Fourier transform of the entire vector function.

I would recommend breaking down the problem into smaller steps and carefully applying the properties of the Fourier transform. It may also be helpful to consult with a colleague or refer to a textbook for guidance.

I hope this helps and good luck with your calculations! Keep pushing through and don't get discouraged by technical difficulties. As scientists, we learn to overcome challenges and find solutions to complex problems.